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Math Expert V
Joined: 02 Sep 2009
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A carpenter constructed a rectangular sandbox with a  [#permalink]

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Difficulty:   5% (low)

Question Stats: 78% (00:50) correct 22% (01:05) wrong based on 1647 sessions

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A carpenter constructed a rectangular sandbox with a capacity of 10 cubic feet. If the carpenter were to make a similar sandbox twice as long, twice as wide, and twice as high as the first sandbox, what would be the capacity, in cubic feet, of the second sandbox?

(A) 20
(B) 40
(C) 60
(D) 80
(E) 100

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Page: 154
Difficulty: 550

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Math Expert V
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Posts: 58410
Re: A carpenter constructed a rectangular sandbox with a  [#permalink]

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8
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SOLUTION

A carpenter constructed a rectangular sandbox with a capacity of 10 cubic feet. If the carpenter were to make a similar sandbox twice as long, twice as wide, and twice as high as the first sandbox, what would be the capacity, in cubic feet, of the second sandbox?

(A) 20
(B) 40
(C) 60
(D) 80
(E) 100

Since given that $$volume=L*W*H=10$$, then $$new \ volume=(2L)*(2W)*(2H)=(2*2*2)*(L*W*H)=8*10=80$$.

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Re: A carpenter constructed a rectangular sandbox with a  [#permalink]

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Let x,y & z be L,W & H
so xyz = 10 cu ft
As per question if we double the lengths on all dimensions
we get (2x)(2y)(2z)=8 xyz= 8*10= 80
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Re: A carpenter constructed a rectangular sandbox with a  [#permalink]

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volume =l*b*h =10 cubic feet
Now we have twice as long, twice as wide, and twice as high as the first sandbox,
L=2*l
B=2*b
H=2*h
final volume =L*B*H =2*l*2*b*2*h=8*(l*b*h) =8*10=80 cubic feet

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Re: A carpenter constructed a rectangular sandbox with a  [#permalink]

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V=LWH

Original Volume = 10

In order to keep things simple. I made Height = 5, Length = 2, and Width = 1

The second statement says double everything.

Height = 10, Length = 4, and Width = 2

V = 80
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Re: A carpenter constructed a rectangular sandbox with a  [#permalink]

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1
2
A quick note on doubling. When you double a length you have 2*L1. When you double all lengths of a rectangle you have (2*L1)(2*L2) = A. An increase of 2^2 or 4. When you double all lengths of a rectangular prism you have (2*L1)(2*L2)(2*L3) = V. An increase of 2^3 or 8.

This leads to the basic relationship:

Line: 2*original size
Rectangle: 4*original size
Rectangular Prism: 8*original size

You can do the math out or memorize this relationship to speed things up.
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Re: A carpenter constructed a rectangular sandbox with a  [#permalink]

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Answer = (D) 80

Original volume = 10 cubic feet

All 3 dimensions made "twice"

New volume = 10 * 2 * 2 * 2 = 80
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Re: A carpenter constructed a rectangular sandbox with a  [#permalink]

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Bunuel wrote:
A carpenter constructed a rectangular sandbox with a capacity of 10 cubic feet. If the carpenter were to make a similar sandbox twice as long, twice as wide, and twice as high as the first sandbox, what would be the capacity, in cubic feet, of the second sandbox?

(A) 20
(B) 40
(C) 60
(D) 80
(E) 100

We are given a rectangular sandbox with a given capacity, which is the volume of the sandbox.

Therefore, we know that the volume of the sandbox is: (L)(W)(H) = 10 cubic feet

We then are told that the carpenter doubles the length, the width, and the height. We can represent this doubling as (2L)(2W)(2H). Thus

(2L)(2W)(2H) = (2)(2)(2)(L)(W)(H) = (2)(2)(2)(10) = 80 cubic feet

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Re: A carpenter constructed a rectangular sandbox with a  [#permalink]

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Using some simple values, let all sides be equal to 2. The first surface area will be 2*2*2 = 8
Now double all these sides. The new surface area will be 4*4*4=64
64 is 8*8. Hence the answer should be 10*8=80 or D
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Re: A carpenter constructed a rectangular sandbox with a  [#permalink]

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I have a small doubt in this question. The volume of a rectangular solid is l*w*h=10*10*10 because the question says its cubic feet. Where am i going wrong in my concept??
Math Expert V
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A carpenter constructed a rectangular sandbox with a  [#permalink]

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longhaul123 wrote:
I have a small doubt in this question. The volume of a rectangular solid is l*w*h=10*10*10 because the question says its cubic feet. Where am i going wrong in my concept??

We are told that the volume is 10 cubic feet (the volume of 3-D objects is measured in cubic units), not that the lengths of the sides are 10 feet, so it should be l*w*h = 10. Please re-read the solutions above.
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Re: A carpenter constructed a rectangular sandbox with a  [#permalink]

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Thank youBunuel.
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Re: A carpenter constructed a rectangular sandbox with a  [#permalink]

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This may is more of a conceptual question I guess. In 2l *2b*2h can we not take the 2 common such as 2(L*B*H).

L*B*H=10.....-> given.

2L*2B*2H= ?
2(L*B*H)
2(10)=20
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Re: A carpenter constructed a rectangular sandbox with a  [#permalink]

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I have a similar question. Why are we not taking the common factor 2(L*B*H)?

Rocket7 wrote:
This may is more of a conceptual question I guess. In 2l *2b*2h can we not take the 2 common such as 2(L*B*H).

L*B*H=10.....-> given.

2L*2B*2H= ?
2(L*B*H)
2(10)=20
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Re: A carpenter constructed a rectangular sandbox with a  [#permalink]

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dixxa wrote:
I have a similar question. Why are we not taking the common factor 2(L*B*H)?

Rocket7 wrote:
This may is more of a conceptual question I guess. In 2l *2b*2h can we not take the 2 common such as 2(L*B*H).

L*B*H=10.....-> given.

2L*2B*2H= ?
2(L*B*H)
2(10)=20

Hi dixxa,
Common factors are to be factored out when the terms are either added or substracted in an equation/expression. In case of multiplication or division, it is simply multiplication or division of terms respectively.

Here, in 2L*2B*2H, we have three terms multplied 2L, 2B, and 2H.

From 2L, 2 can be factored out. (you know 2L=2 multiplied by L)

Similarly, From 2B, 2 can be factored out.
Similarly, From 2H, 2 can be factored out.

Now, 2L*2B*2H=2*2*2*L*B*H=8(L*B*H)

Had it been 2L+2B+2H, then we can factor out 2 as '2' is a common factor of 2L, 2B, and 2H, making 2L+2B+2H=2(L+B+H)

Hope it clarifies your query.
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Intern  B
Joined: 15 Nov 2013
Posts: 3
Re: A carpenter constructed a rectangular sandbox with a  [#permalink]

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Thank you so much for your explanation. It makes sense.

PKN wrote:
dixxa wrote:
I have a similar question. Why are we not taking the common factor 2(L*B*H)?

Rocket7 wrote:
This may is more of a conceptual question I guess. In 2l *2b*2h can we not take the 2 common such as 2(L*B*H).

L*B*H=10.....-> given.

2L*2B*2H= ?
2(L*B*H)
2(10)=20

Hi dixxa,
Common factors are to be factored out when the terms are either added or substracted in an equation/expression. In case of multiplication or division, it is simply multiplication or division of terms respectively.

Here, in 2L*2B*2H, we have three terms multplied 2L, 2B, and 2H.

From 2L, 2 can be factored out. (you know 2L=2 multiplied by L)

Similarly, From 2B, 2 can be factored out.
Similarly, From 2H, 2 can be factored out.

Now, 2L*2B*2H=2*2*2*L*B*H=8(L*B*H)

Had it been 2L+2B+2H, then we can factor out 2 as '2' is a common factor of 2L, 2B, and 2H, making 2L+2B+2H=2(L+B+H)

Hope it clarifies your query.
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Re: A carpenter constructed a rectangular sandbox with a  [#permalink]

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Bunuel wrote:
A carpenter constructed a rectangular sandbox with a capacity of 10 cubic feet. If the carpenter were to make a similar sandbox twice as long, twice as wide, and twice as high as the first sandbox, what would be the capacity, in cubic feet, of the second sandbox?

(A) 20
(B) 40
(C) 60
(D) 80
(E) 100

Practice Questions
Question: 18
Page: 154
Difficulty: 550

If each of the sides was increased by a factor of 2,

then the linear ratio is 1:2. In order to get the volume ratio, I need to

cube the linear ratio; so the volume ratio is going to be (1 cubed), to (2 cubed) -- which is 8.

So if this is 10 cubic feet, this is going to be 80 cubic feet, and the correct answer choice is D.
_________________ Re: A carpenter constructed a rectangular sandbox with a   [#permalink] 23 Jun 2019, 06:28
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